Many-body localization in Fock-space: a local perspective

TL;DR

This paper investigates many-body localization in a disordered fermionic system by analyzing self-energy distributions on Fock-space, combining mean-field theory and numerical exact diagonalization to identify transition signatures.

Contribution

It introduces a local perspective on MBL using self-energy distributions on Fock-space and develops a probabilistic mean-field theory validated by numerical results.

Findings

MBL transition captured by self-energy distribution analysis

Universal long-tailed Levy behavior in MBL phase

Log-normal distributions in delocalized phase

Abstract

A canonical model for many-body localization (MBL) is studied, of interacting spinless fermions on a lattice with uncorrelated quenched site-disorder. The model maps onto a tight-binding model on a `Fock-space (FS) lattice' of many-body states, with an extensive local connectivity. We seek to understand some aspects of MBL from this perspective, via local propagators for the FS lattice and their self-energies (SE's); focusing on the SE probability distributions, over disorder and FS sites. A probabilistic mean-field theory (MFT) is first developed, centered on self-consistent determination of the geometric mean of the distribution. Despite its simplicity this captures some key features of the problem, including recovery of an MBL transition, and predictions for the forms of the SE distributions. The problem is then studied numerically in $1d$ by exact diagonalization, free from MFT assumptions. The geometric mean indeed appears to act as a suitable order parameter for the transition. Throughout the MBL phase the appropriate SE distribution is confirmed to have a universal form, with long-tailed L\'evy behavior as predicted by MFT. In the delocalized phase for weak disorder, SE distributions are clearly log-normal; while on approaching the transition they acquire an intermediate L\'evy-tail regime, indicative of the incipient MBL phase.